Package 'cctools'

Tools for the continuous convolution trick in nonparametric estimation

Description

Implements the uniform scaled beta distribution dusb(), a generic function for continuous convolution cont_conv(), and the continuous convolution kernel density estimator cckde().

Author(s)

Thomas Nagler

References

Nagler, T. (2017). A generic approach to nonparametric function estimation with mixed data. arXiv:1704.07457

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Continuous convolution density estimator

Description

The continuous convolution kernel density estimator is defined as the classical kernel density estimator based on continuously convoluted data (see cont_conv()). cckde() fits the estimator (including bandwidth selection), dcckde() and predict.cckde() can be used to evaluate the estimator.

Usage

cckde(x, bw = NULL, mult = 1, theta = 0, nu = 5, ...)

dcckde(x, object)

## S3 method for class 'cckde'
predict(object, newdata, ...)

Arguments

x	a matrix or data frame containing the data (or evaluation points).
bw	vector of bandwidth parameter; if NULL, the bandwidths are selected automatically by likelihood cross validation.
mult	bandwidth multiplier; either a positive number or a vector of such. Each bandwidth parameter is multiplied with the corresponding multiplier.
theta	scale parameter of the USB distribution (see, dusb()).
nu	smoothness parameter of the USB distribution (see, dusb()). The estimator uses the Epanechnikov kernel for smoothing and the USB distribution for continuous convolution (default parameters correspond to the uniform distribution on $[-0.5, 0.5]$.
...	unused.
object	cckde object.
newdata	matrix or data frame containing evaluation points.

Details

If a variable should be treated as ordered discrete, declare it as ordered(), factors are expanded into discrete dummy codings.

References

Nagler, T. (2017). A generic approach to nonparametric function estimation with mixed data. arXiv:1704.07457

Examples

# dummy data with discrete variables
dat <- data.frame(
    F1 = factor(rbinom(10, 4, 0.1), 0:4),
    Z1 = ordered(rbinom(10, 5, 0.5), 0:5),
    Z2 = ordered(rpois(10, 1), 0:10),
    X1 = rnorm(10),
    X2 = rexp(10)
)

fit <- cckde(dat)  # fit estimator
dcckde(dat, fit)   # evaluate density
predict(fit, dat)  # equivalent

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Continuous convolution

Description

Applies the continuous convolution trick, i.e. adding continuous noise to all discrete variables. If a variable should be treated as discrete, declare it as ordered() (passed to expand_as_numeric()).

Usage

cont_conv(x, theta = 0, nu = 5, quasi = TRUE)

Arguments

x	data; numeric matrix or data frame.
theta	scale parameter of the USB distribution (see, dusb()).
nu	smoothness parameter of the USB distribution (see, dusb()). The estimator uses the Epanechnikov kernel for smoothing and the USB for continuous convolution (default parameters correspond to the $U[-0.5, 0.5]$ distribution).
quasi	logical indicating whether quasi random numbers sholuld be used (qrng::ghalton()); only works for theta = 0.

Details

The UPSB distribution (dusb()) is used as the noise distribution. Discrete variables are assumed to be integer-valued.

Value

A data frame with noise added to each discrete variable (ordered columns).

References

Nagler, T. (2017). A generic approach to nonparametric function estimation with mixed data. arXiv:1704.07457

Examples

# dummy data with discrete variables
dat <- data.frame(
    F1 = factor(rbinom(10, 4, 0.1), 0:4),
    Z1 = ordered(rbinom(10, 5, 0.5), 0:5),
    Z2 = ordered(rpois(10, 1), 0:10),
    X1 = rnorm(10),
    X2 = rexp(10)
)

pairs(dat)
pairs(expand_as_numeric(dat))  # expanded variables without noise
pairs(cont_conv(dat))          # continuously convoluted data

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Uniform scaled beta distribution

Description

The uniform scaled beta (USB) distribution describes the distribution of the random variable

$U_{b, \nu} = U + \theta(B - 0.5),$

where $U$ is a $U[-0.5, 0.5]$ random variable, $B$ is a $Beta(\nu, \nu)$ random variable, and $theta > 0, \nu >= 1$.

Usage

dusb(x, theta = 0, nu = 5)

rusb(n, theta = 0, nu = 5, quasi = FALSE)

Arguments

x	vector of quantiles.
theta	scale parameter of the USB distribution.
nu	smoothness parameter of the USB distribution.
n	number of observations.
quasi	logical indicating whether quasi random numbers (qrng::ghalton()) should be used for generating uniforms (which are then transformed by the quantile function)

References

Nagler, T. (2017). A generic approach to nonparametric function estimation with mixed data. arXiv:1704.07457

Examples

# plot distribution
sq <- seq(-0.8, 0.8, by = 0.01)
plot(sq, dusb(sq), type = "l")
lines(sq, dusb(sq, theta = 0.25), col = 2)
lines(sq, dusb(sq, theta = 0.25, nu = 10), col = 3)

# simulate from the distribution
x <- rusb(100, theta = 0.3, nu = 0)

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Numeric model matrix for continuous convolution

Description

Turns ordered variables into integers and expands factors as binary dummy codes. cont_conv() additionally adds noise to discrete variables, but this is only useful for estimation. [cc_prepare()] can be used to evaluate an already fitted estimate.

Usage

expand_as_numeric(x)

Arguments

x	a vector or data frame with numeric, ordered, or factor columns.

Value

A numeric matrix containing the expanded variables. It has additional type expanded_as_numeric and attr(, "i_disc") cntains the indices of discrete variables.

Examples

# dummy data with discrete variables
dat <- data.frame(
    F1 = factor(rbinom(100, 4, 0.1), 0:4),
    Z1 = as.ordered(rbinom(100, 5, 0.5)),
    Z2 = as.ordered(rpois(100, 1)),
    X1 = rnorm(100),
    X2 = rexp(100)
)

pairs(dat)
pairs(expand_as_numeric(dat))  # expanded variables without noise
pairs(cont_conv(dat))          # continuously convoluted data

---

Expands names for expand_as_numeric

Description

Expands each element according to the factor expansions of columns in expand_as_numeric().

Usage

expand_names(x)

Arguments

x	as in expand_as_numeric().

Value

A vector of size ncol(expand_as_numeric(x)).

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Expand a vector like expand_as_numeric

Description

Expands each element according to the factor expansions of columns in expand_as_numeric().

Usage

expand_vec(y, x)

Arguments

y	a vector of length 1 or ncol(x).
x	as in expand_as_numeric().

Value

A vector of size ncol(expand_as_numeric(x)).

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